Indices

The Pattern of Decreasing Powers

Recall that a positive index represents repeated multiplication:
$2^3 = 8, \quad 2^2 = 4, \quad 2^1 = 2$

Notice what happens as the index decreases by $1$: we divide by the base ($2$) at each step.

If we continue this pattern beyond $2^1$:

• $2^0 = 2 \div 2 = 1$
• $2^{-1} = 1 \div 2 = \frac{1}{2}$
• $2^{-2} = \frac{1}{2} \div 2 = \frac{1}{4} = \frac{1}{2^2}$

Extending this logic works for any non-zero base, giving us a natural bridge to zero and negative indices!

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Visualizing Powers of a Base

Here is how decreasing exponents move us from positive powers to fractions:

Key Takeaways:

• Decreasing an index by $1$ means dividing by the base.
• The zero power acts as the anchor point ($a^0 = 1$).
• Negative powers generate reciprocals, not negative numbers.

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The Zero Index Rule

We can also prove why $a^0 = 1$ using the division rule for indices: $\frac{a^m}{a^n} = a^{m-n}$.

Consider dividing any non-zero expression by itself:
$\frac{a^n}{a^n} = 1$

Using the quotient rule on the exact same expression:
$\frac{a^n}{a^n} = a^{n-n} = a^0$

Equating the two results:
$a^0 = 1 \quad (a \neq 0)$

• Rule: Any non-zero base raised to the power of $0$ equals $1$.
• Examples: $7^0 = 1$, $(-5)^0 = 1$, $(100x)^0 = 1$.
• Note: $0^0$ is undefined because division by zero is not defined.

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The Negative Index Rule

Using the same division rule, consider $\frac{a^0}{a^n}$:

Using the quotient rule:
$\frac{a^0}{a^n} = a^{0-n} = a^{-n}$

Since $a^0 = 1$, we can substitute $1$ into the numerator:
$\frac{a^0}{a^n} = \frac{1}{a^n}$

Comparing the two forms gives our main rule:
$a^{-n} = \frac{1}{a^n} \quad (a \neq 0)$

Useful extensions:

• $\dfrac{1}{a^{-n}} = a^n$ (a negative power in the denominator moves to the numerator as positive)
• $\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$ (flip the fraction to make the power positive)

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Worked Example: Problem Overview

Problem: Simplify the expression $\dfrac{6x^{-3}y^4}{2x^2y^{-2}}$ and write your answer using only positive indices.

To solve this, we track how factors with negative indices relocate across the fraction bar:

We will separate numerical coefficients from variable terms step by step.

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Worked Example: Step-by-Step Solution

Step 1: Divide numerical coefficients.
$\frac{6}{2} = 3$

Step 2: Apply the negative index rule to relocate factors.
$\frac{6x^{-3}y^4}{2x^2y^{-2}} = 3 \cdot \frac{y^4 \cdot y^2}{x^2 \cdot x^3}$

Step 3: Combine powers with the same base using $a^m \cdot a^n = a^{m+n}$.
$= \frac{3 y^{4+2}}{x^{2+3}} = \frac{3y^6}{x^5}$

Common Pitfall: Do not make coefficients negative! A negative power like $x^{-3}$ means reciprocal ($\frac{1}{x^3}$); it does not change $+6$ into a negative number.